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  "body": "\n## Summary\n\nThe hypergeometric differential equation is the linear equation $z(1-z)w'' + [c - (a+b+1)z]w' - ab\\,w = 0$.\nIt is a [[Fuchsian equation|ZBVH2YE50B]] with exactly three [[regular singular points|H96HXG6CA0]], at $0$, $1$ and $\\infty$.\nThe [[hypergeometric function|463X5V31D9]] is its solution that is holomorphic at $0$ with value $1$ there.\nEvery second-order Fuchsian equation with three singular points reduces to this form.\nOne uses a Möbius change of variable and a change of the unknown function.\n\n## Explanations\n\n### L1 human\n\nMany equations of physics, after separating variables, become second-order linear equations whose coefficients blow up at a few points.\nThe hypergeometric equation is the simplest interesting case: it has three such points, $0$, $1$ and $\\infty$, and near each of them solutions behave like powers.\nSolving it by a power series around $0$ gives the hypergeometric series.\nFor example, with $a = 1$, $b = c$ the equation is solved by $(1 - z)^{-1}$, which is the geometric series.\n\n### L2 human\n\nThe local exponents are $0, 1 - c$ at $z = 0$; $0, c - a - b$ at $z = 1$; and $a, b$ at $z = \\infty$.\nTheir sum is $1$, as Fuchs's relation requires for three singular points.\nStudying how the solutions change around the singular points leads to the [[monodromy of the hypergeometric equation|T8SBE28DM1]].\n",
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